When \(M\)  is a three dimensional ball with multiple circular absorbing windows on the boundary, an expansion
capturing the explicit form of the \(O(1)\)  correction in terms of the Neumann Green's function and its regular part was done
in [1]}. The method of matched asymptotic used there required the explicit computation of the Neumann Green's function, which is only possible in special geometries with high degrees of symmetry/homogeneity. In these results one does not see the full effects of local geometry. This result was also rigorously proved in [2]} but with a better estimate for the error term.
